Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x² - x - 6 < 0

Verified step by step guidance

1. Start by rewriting the inequality: \(x^{2} - x - 6 < 0\).
2. Factor the quadratic expression on the left side. Find two numbers that multiply to \(-6\) and add to \(-1\). This gives the factors: \((x - 3)(x + 2)\).
3. Set each factor equal to zero to find the critical points: \(x - 3 = 0\) and \(x + 2 = 0\), which gives \(x = 3\) and \(x = -2\).
4. Use the critical points to divide the number line into three intervals: \((-\infty, -2)\), \((-2, 3)\), and \((3, \infty)\). Test a value from each interval in the inequality \((x - 3)(x + 2) < 0\) to determine where the product is negative.
5. Based on the test results, write the solution set in interval notation, including only the intervals where the inequality holds true.

Key Concepts

Solving Quadratic Inequalities

Solving quadratic inequalities involves finding the values of the variable that make the quadratic expression less than, greater than, or equal to zero. This typically requires factoring the quadratic, identifying critical points, and testing intervals to determine where the inequality holds true.

Factoring Quadratic Expressions

Factoring is the process of expressing a quadratic polynomial as a product of two binomials. For example, x² - x - 6 factors into (x - 3)(x + 2). Factoring helps find the roots or zeros of the quadratic, which are essential for determining the intervals to test in inequalities.

Interval Notation

Interval notation is a concise way to represent sets of numbers, especially solutions to inequalities. It uses parentheses and brackets to indicate open or closed intervals, respectively. For example, (−2, 3) represents all numbers between -2 and 3, not including the endpoints.